Integral of \( \displaystyle \sqrt{x + 1} \)
Problem 4.797 · easy
Find \( \displaystyle \int \sqrt{x + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \sqrt{x + 1}\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \]antiderivativeApply the power rule for integration.✓ Proved
Answer \( \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} + C \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the power rule for integration via substitution (implicitly handled by the single-step antiderivative rule for linear arguments) and simplifies the result. Each step changes only one aspect of the expression and uses valid labels.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the power rule for integration via substitution (implicitly handled by the single-step antiderivative rule for linear arguments) and simplifies the result. Each step changes only one aspect of the expression and uses valid labels.qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 3 applies the power rule for integration to (x+1)^(1/2) but fails to account for the chain rule factor (the derivative of the inner function x+gpt-oss:20b: pass 2026-10-10
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-10 with SymPy 1.14.0.